LE^{t}_{ \to } , LR^{ \circ }_{{\widehat{ \sim }}}, LK and Cutfree Proofs

Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique—attributable to Gentzen—that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen’s sequent calculus $ LK $ without any use of mix.     

The representation of Takeuti's\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} -operator

Gaisi Takeuti has recently proposed a new operation on orthomodular latticesL, \(\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} \) :P(L)»L. The properties of \(\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} \) suggest that the value of \(\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} \) (A) (A) \( \subseteq \) L) corresponds to the degree in which the elements ofA behave classically. To make this idea precise, we investigate the connection between structural properties of orthomodular latticesL and the existence of two-valued homomorphisms onL.     

Categorical Equivalence Between $$\varvec{PMV}_{\varvec{f}}$$ PMV f -Product Algebras and Semi-Low $$\varvec{f}_{\varvec{u}}$$ f u -Rings

Studia Logica - An explicit categorical equivalence is defined between a proper subvariety of the class of $${ PMV}$$ -algebras, as defined by Di Nola and Dvure?enskij, to be called $${...     

Arithmetical Soundness and Completeness for $$\varvec{\Sigma }_{\varvec{2}}$$ Numerations

We prove that for each recursively axiomatized consistent extension T of Peano Arithmetic and \(n \ge 2\), there exists a \(\Sigma _2\) numeration \(\tau (u)\) of T such that the provability logic of the provability predicate \(\mathsf{Pr}_\tau (x)\) naturally constructed from \(\tau (u)\) is exactly \(\mathsf{K}+ \Box (\Box ^n p \rightarrow p) \rightarrow \Box p\). This settles Sacchetti’s problem affirmatively.     

Types of \mathsf{I} -Free Hereditary Right Maximal Terms

The implicational fragment of the relevance logic “ticket entailment” is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of T_→ are in normal form and built in all but one case from and only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define FIT_→ – a Fitch-style calculus (related to FT_→) for the implicational fragment of ticket entailment.     

$$\mathrm {ZF}$$ ZF Between Classicality and Non-classicality

Studia Logica - We present a generalization of the algebra-valued models of $$\mathrm {ZF}$$ where the axioms of set theory are not necessarily mapped to the top element of an algebra, but may get...     

{ }-{ } Double twinning in magnesium

Microstructural investigations have been carried out on a fully lamellar Ti₄₉Al₄₇Cr₂Nb₂     

Analyticity,Balance and Non-admissibility of $$\varvec{Cut}$$ Cut in Stoic Logic

Studia Logica - This paper shows that, for the Hertz–Gentzen Systems of 1933 (without Thinning), extended by a classical rule T1 (from the Stoics) and using certain axioms (also from the...     

Two Maximality Results for the Lattice of Extensions of $$\vdash _{\mathbf {RM}}$$

Studia Logica - We use an algebraic argument to prove that there are exactly two premaximal extensions of $$\mathbf {RM}$$ ’s consequence. We also show that one of these extensions is the...     

An Abductive Question-Answer System for the Minimal Logic of Formal Inconsistency $$\mathsf {mbC}$$ mbC

Studia Logica - The aim in this paper is to define an Abductive Question-Answer System for the minimal logic of formal inconsistency $$\mathsf {mbC}$$ . As a proof-theoretical basis we employ the...     

Estimating the Cognitive Diagnosis $$\varvec{Q}$$ Q Matrix with Expert Knowledge: Application to the Fraction-Subtraction Dataset

Psychometrika - Cognitive diagnosis models (CDMs) are an important psychometric framework for classifying students in terms of attribute and/or skill mastery. The $$\varvec{Q}$$ matrix, which...     

Antiphase domains and {011} twins in TiAl

Abstract

Titanium aluminides with a TiAl/Ti₃Al two-phase microstructure exhibit mechanical properties which are superior to those for either single-phase TiAl or Ti₃Al. The TiAl lamella in the two-phase microstructures show differently oriented TiAl regions which are frequently observed to coexist within the same TiAl lamella. The atomic arrangements of these regions are similar to antiphase domains/{011} twins of TiAl. However, the mechanism by which these regions are formed cannot be derived from the TiAl phase alone. This study suggests that, for two-phase titanium aluminides, Ti₃Al has a strong influence on the development of the TiAl microstructure.     

The Weak Normalization of the Simply Typed {lambda}se-calculus

In this paper, we show the weak normalization (WN) of the simply-typeds_e-calculus with open terms where abstractions are decoratedwith types, and metavariables, de Bruijn indices and updatingoperators are decorated with environments. We show a proof ofWN using the _e-calculus, a calculus isomorphic to . This proof is strongly influenced by Goubault-Larrecq'sproof of WN for the -calculus but with subtle differences whichshow that the two styles require different attention. Furthermore,we give a new calculus '_e which works like s_e but which iscloser to than _e. For both _e and '_e we prove WN for typedsemi-open terms (i.e. those which allow term variables but nosubstitution variables), unlike the result of Goubault-Larrecqwhich covered all open terms.